Independent of their usage, numbers become fascinating entities, and their mathematical relationships express the complexity of a vast system underpinning nature itself. The likes of 6, 28 and 496 turn into something more sublime than simple carriers of information. They look at the world of numbers and, just as you'd never define your human beloved solely by his or her profession or hair color, the math lover sees beyond the mere function of numbers. In the eyes of some, there is no finer beauty than that found in mathematics. They trace the undercurrents of our personality, and, to the observant and loving eye, they illuminate true beauty. Such interrelated details come to define us. The way he or she lets old paperbacks stack up on the bedside table. The peculiarities of the other person's morning coffee ritual. The silly in-jokes shared at the end of the day. The course can serve as a foundation for anyone wishing to pursue research involving function fields, such as arithmetic on algebraic curves, zeta functions of curves, curve cryptography, algebraic geometry codes, Drinfeld modules and other topics.Andrea Pistolesi/The Image Bank/ Getty ImagesĪnyone who has ever fallen in love will tell you it's the little things about the other person that matter. Topics will include some or all of the following: valuation theory, algebraic function fi elds, divisors, extensions of function fields, class groups, elliptic and hyperelliptic curves and their function fields. This course represents a basic introduction to function fields and algebraic curves over finite fields from a number theoretic perspective. This allows us to provide these very powerful tools to students at an early stage and hope that one day they can discover many more beautiful applications of these tools.Ī n Introduction to Global Function Fields
Although this limits the strength of the theorems and the scope of their application, the trade off is that only a minimal prerequisite is required. The goal of this course is to introduce versions of these theorems involving only absolute values over the rational numbers. Without the Subspace Theorem, several elementary-to-state yet surprisingly-hard-to-prove results would have remained open. Roth's Theorem and especially its generalization, the Subspace Theorem, are among the milestones of diophantine geometry in the second half of the 20th century. Roth's Theorem, Subspace Theorem, and Some Applications Some basic notions on Fourier analysis or complex analysis are preferable (background material shall be provided). We invite the participants to discover a few notions regarding the zeros of the Riemann zeta function as well as some analytical tools to deduce the prime number theorem. First conjectured by Gauss, then proven by de la Vallée Poussin and Hadamard, building upon some groundbreaking ideas of Dirichlet and Riemann, this result is seminal in the field of analytic number theory. The Prime Number Theorem gives an estimate for the number of primes pi(x) up to an asymptotically large number x. Students from under-represented groups or those committed to EDI are encouraged to apply. In exemplary cases, upper-year undergraduate students will also be considered.
The summer school is * free* and open to MSc students from a Canadian or North-West American University. The goal of this event is to promote positive cooperation and collaboration amongst students, while bringing light to equity, diversity, and inclusion issues within the study of Mathematics. This summer school will introduce students to Number Theory topics being researched throughout Alberta: algebraic number theory, diophantine geometry, and analytic number theory.